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We consider the problem of numerical integration for weighted anchored andANOVA Sobolev spaces of $s$-variate functions. Here $s$ is large including$s=\infty$. Under the assumption of sufficiently fast decaying weights, weprove in a constructive way that such integrals can be approximated byquadratures for functions $f_k$ with only $k$ variables, where$k=k(\varepsilon)$ depends solely on the error demand $\varepsilon$ and issurprisingly small when $s$ is sufficiently large relative to $\varepsilon$.This holds, in particular, for $s=\infty$ and arbitrary $\varepsilon$ sincethen $k(\varepsilon)<\infty$ for all $\varepsilon$. Moreover $k(\varepsilon)$does not depend on the function being integrated, i.e., is the same for allfunctions from the unit ball of the space.

Very Low Truncation Dimension for High Dimensional Integration Under Modest Error Demand